Author

Theresa Szczepanski

Published

October 22, 2023

Code 
source('dependencies.R')
knitr::opts_chunk$set(echo = TRUE, warning=FALSE, message=FALSE)
#install.packages("stargazer")
library(stargazer)

Research Questions

The Massachusetts Education Reform Act in 1993 was passed in the context of a national movement toward education reform throughout the United States. As early as 1989 there were calls to establish national curriculum standards as a way to improve student college and career readiness skills and close poverty gaps (Greer 2018). Massachusetts Comprehensive Assessment System (MCAS) tests were introduced as part of the Massachusetts Education Reform Act.

The MCAS tests are a significant tool for educational equity. Scores on the Grade 10 Math MCAS test “predict longer-term educational attainments and labor market success, above and beyond typical markers of student advantage” and differences among students are largely and “sometimes completely accounted for” by differences in 10th grade MCAS scores and educational attainments. (Papy 2020).

With the introduction of the new Common Core standards and accountability testing came the demand for aligned curricular materials and teaching practices. Research indicates that the choice of instructional materials can have an impact “as large as or larger than the impact of teacher quality” (Chingos 2012). Massachusetts, along with Arkansas, Delaware, Kentucky, Louisiana, Maryland, Mississippi, Nebraska, New Mexico, Ohio, Rhode Island, Tennessee, and Texas belongs to the Council of Chief State School Officers’ (CCSO), High Quality Instructional Materials and Professional Development network which aims to close the “opportunity gap” among students by ensuring that every teacher has access to high-quality, standards aligned instructional materials and receives relevant professional development to support their use of these materials (Chief State School Officers 2021).

All Massachusetts Public School students must complete a High School science MCAS exam providing a wealth of standardized data on students’ discipline specific skill development. All schools receive annual summary reports on student performance. Significant work has been done using the MCAS achievement data and the Student Opportunity Act to identify achievement gaps and address funding inequities across the Commonwealth (Papy 2020). With funding gaps outlined in the late 1990’s closing, one could consider how the MCAS data could be leveraged to support the state’s current high quality instructional materials initiatives. The state compiles school’s performance disaggregated by each MCAS question item (DESE 2022).

Using the curricular information provided in state wide Next Generation MCAS High School Introductory Physics Item reports together with school-level student performance data, we hope to address the following broad questions:

  • Is there a relationship between differences in a school’s performance across Science Practice Categories and a school’s overall achievement on the Introductory Physics exam?

  • How can trends in a school’s performance be used to provide schools with guidance on discipline-specific curricular areas to target to improve student achievement?

In this report, I will analyze the High School Introductory Physics Next Generation Massachusetts Comprehensive Assessment System (MCAS) tests results for Massachusetts public schools.

Data for the study were drawn from DESE’s Next Generation MCAS Test Achievement Results statewide report, Item Analysis statewide report, and the MCAS digital item library. The Next Generation High School Introductory Physics MCAS assessment consists of 42 multiple choice and constructed response items that assess students on Physical Science standards from the 2016 STE Massachusetts Curriculum Framework in the content Reporting Categories of Motions and Forces, MF, Energy, EN, and Waves, WA. Each item is associated with a specific content standard from the Massachusetts Curriculum Framework as well as an underlying science Practice Category of Evidence Reasoning and Modeling, ERM, Mathematics and Data, MD, or Investigations and Questioning, IQ. The State Item Report provides the percentage of points earned by students in a school for each item as well as the percentage of points earned by all students in the state for each item.

The HSPhy_NextGen_SchoolSum data frame contains summary performance results from 112 public schools across the commonwealth on the Next Generation High School Introductory Physics MCAS, which was administered in the Spring of 2022 and 2023. 87 schools tested students in both years and 25 schools only tested students in 1 of the 2 testing years, with 27,745 students completing the exam.

For each school, there are values reported for 44 different variables which consist of information from three broad categories

  • School Characteristics: This includes the name of the school and the size of the school, School Size, as determined by the number of students that completed the MCAS exam.

  • Discipline-Specifc Performance Metrics: This includes the percentage of points earned by students at a school for items each content Reporting Category, MF%, EN%, WA% and science Practice Category ERM%, MD%, IQ%, the difference between a school’s percentage of points earned compared to the percentage of points earned by all students in the state (MFDiff, ENDiff, etc…), and the variability in a school’s performance relative to the state by category as measured by the standard deviation of the school’s Diff across categories (SD MF Diff, SD EN Diff, etc…).

  • Aggregate Performance Level metrics: This includes a school’s percentage of students at each of the four Performance Levels, (E%: Exceeding Expectations, M%: Meeting Expectations, PM%: Partially Meeting Expectations, and NM%: Not Meeting Expectations), the difference between these percentages and the percentage of students in Massachusetts at each performance level (EDiff, MDiff, PMDiff, NMDiff), and an ordinal classification of school’s, EM Perf Stat based on the percentage of students that were classified as Exceeding or Meeting expectations on the exam (HighEM, HighM, Mid, Mid-Low, Low).

See the HSPhy_NextGenMCASDF data frame summary and codebook for further details about all variables.

Hypothesis

  • A school’s percentage of students classified as Exceeding expectations on the Introductory Physics MCAS is negatively associated with a school’s variance in performance relative to students in the state on Mathematics and Data items, SD MD Diff.

  • A school’s summary performance on items in a given content Reporting Category as measured by MF%, EN%, and WA%, is positively associated with the Reporting Category's weight within the exam.

Descriptive Statistics

Code 
#HSPhy_NextGen_SchoolSum
HSPhy_NextGen_SchoolSum<-HSPhy_NextGen_SchoolSum%>%
  ungroup()

#HSPhy_NextGen_SchoolSum
# HSPhy_NextGen_PerfDF
# HSPhy_NextGen_SchoolIT301DF

HSPhy_2023_SchoolSizeDF<-read_excel("data/2023_Physics_NextGenMCASItem.xlsx", skip = 1)%>%
  select(`School Name`, `School Code`, `Tested`)%>%
  mutate(`Tested` = as.integer(`Tested`))%>%
  select(`School Name`, `School Code`, `Tested`)

HSPhy_2022_SchoolSizeDF<-read_excel("data/2022_Physics_NextGenMCASItem.xlsx", skip = 1)%>%
  select(`School Name`, `School Code`, `Tested`)%>%
  mutate(`Tested` = as.integer(`Tested`))%>%
  select(`School Name`, `School Code`, `Tested`)


HSPhy_SchoolSize <- rbind(HSPhy_2023_SchoolSizeDF, HSPhy_2022_SchoolSizeDF)%>%
  mutate(count = 1)%>%
  group_by(`School Name`, `School Code`)%>%
  summarise(count = sum(count),
            `Tested` = sum(`Tested`))%>%
  mutate(`Tested Count` = round(`Tested`/count))%>%
  ungroup()
#HSPhy_SchoolSize
quantile <- quantile(HSPhy_SchoolSize$`Tested Count`)
HSPhy_Size<-HSPhy_SchoolSize%>%
  mutate(`School Size` = case_when(
    `Tested Count` <= quantile[2] ~ "Small",
    `Tested Count` > quantile[2] &
      `Tested Count` <= quantile[3] ~ "Low-Mid",
    `Tested Count` > quantile[3] &
      `Tested Count` <= quantile[4] ~ "Upper-Mid",
    `Tested Count` > quantile[4] &
      `Tested Count` <= quantile[5] ~ "Large",
  ))%>%
  mutate(`School Size` = recode_factor(`School Size`,
                                            "Small" = "Small",
                                            "Low-Mid" = "Low-Mid",
                                            "Upper-Mid" = "Upper-Mid",
                                            "Large" = "Large",
                                            .ordered = TRUE))%>%
  select(`School Name`, `School Code`, `School Size`)


#HSPhy_Size

HSPhy_NextGen_SchoolSum<-HSPhy_NextGen_SchoolSum%>%
  left_join(HSPhy_Size, by = c("School Name" = "School Name", "School Code" = "School Code"))%>%
  mutate(`EMDiff` = `EDiff` + `MDiff`)%>%
  mutate(`EM Perf Stat` = case_when(
    `EDiff` > 0 & `EDiff` + `MDiff` > 0 ~ "HighEM",
    `EDiff` <= 0 & `EDiff` + `MDiff` > 0 ~ "HighM",
    #`EMDiff` > quantile(HSPhy_NextGen_SchoolSum$`EMDiff`)[3] & 
      `EMDiff` <= 0  & `EMDiff` > -14 ~ "Mid",
    `EMDiff` <= -14 & `EMDiff` >= -33  ~ "Mid-Low",
    `EMDiff` < -33  ~ "Low"
   
  ))%>%
  mutate(`EM Perf Stat` = recode_factor(`EM Perf Stat`,
                                 "HighEM" = "HighEM",
                                 "HighM" = "HighM",
                                 "Mid" = "Mid",
                                 "Mid-Low" = "Mid-Low",
                                 "Low" = "Low",
                                 .ordered = TRUE))
HSPhy_NextGen_SchoolSum
Code 
#quantile(HSPhy_NextGen_SchoolSum$`EMDiff`)

                                      

#summary(HSPhy_NextGen_SchoolSum)
print(summarytools::dfSummary(HSPhy_NextGen_SchoolSum,
                         varnumbers = FALSE,
                         plain.ascii  = FALSE,
                         style        = "grid",
                         graph.magnif = 0.70,
                        valid.col    = FALSE),
       method = 'render',
       table.classes = 'table-condensed')

Data Frame Summary

HSPhy_NextGen_SchoolSum

Dimensions: 112 x 40
Duplicates: 0
Variable Stats / Values Freqs (% of Valid) Graph Missing
Subject [character] 1. PHY
112 ( 100.0% )
0 (0.0%)
School Name [character]
1. Academy Of the Pacific Ri
2. Agawam High
3. Andover High
4. Another Course To College
5. Arlington High
6. Assabet Valley Vocational
7. Athol High
8. Atlantis Charter School
9. Attleboro High
10. B M C Durfee High
[ 102 others ]
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
102 ( 91.1% )
0 (0.0%)
School Code [character]
1. 00050505
2. 00090505
3. 00100505
4. 00160505
5. 00200505
6. 00230505
7. 00260505
8. 00350426
9. 00350505
10. 00350507
[ 102 others ]
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
1 ( 0.9% )
102 ( 91.1% )
0 (0.0%)
EN% [numeric]
Mean (sd) : 47.7 (14)
min ≤ med ≤ max:
19 ≤ 47 ≤ 80
IQR (CV) : 19.5 (0.3)
50 distinct values 0 (0.0%)
MF% [numeric]
Mean (sd) : 52.2 (13.5)
min ≤ med ≤ max:
22 ≤ 53 ≤ 79
IQR (CV) : 18.2 (0.3)
48 distinct values 0 (0.0%)
WA% [numeric]
Mean (sd) : 45 (13.1)
min ≤ med ≤ max:
20 ≤ 44 ≤ 79
IQR (CV) : 17.2 (0.3)
47 distinct values 0 (0.0%)
EN Diff SD [numeric]
Mean (sd) : 8.5 (3.4)
min ≤ med ≤ max:
2.5 ≤ 7.6 ≤ 20
IQR (CV) : 4.2 (0.4)
108 distinct values 0 (0.0%)
MF Diff SD [numeric]
Mean (sd) : 8.7 (3.1)
min ≤ med ≤ max:
3.4 ≤ 8.3 ≤ 20.1
IQR (CV) : 4.1 (0.4)
106 distinct values 0 (0.0%)
WA Diff SD [numeric]
Mean (sd) : 8.3 (2.9)
min ≤ med ≤ max:
2.9 ≤ 7.8 ≤ 15.2
IQR (CV) : 3.6 (0.3)
106 distinct values 0 (0.0%)
IQ% [numeric]
Mean (sd) : 49.4 (16.4)
min ≤ med ≤ max:
3 ≤ 50 ≤ 86
IQR (CV) : 24.2 (0.3)
55 distinct values 0 (0.0%)
MD% [numeric]
Mean (sd) : 47.2 (14.4)
min ≤ med ≤ max:
17 ≤ 47 ≤ 80
IQR (CV) : 21.2 (0.3)
48 distinct values 0 (0.0%)
ERM% [numeric]
Mean (sd) : 53 (12.6)
min ≤ med ≤ max:
26 ≤ 52.5 ≤ 80
IQR (CV) : 17.2 (0.2)
49 distinct values 0 (0.0%)
None% [numeric]
Mean (sd) : 45.4 (13.1)
min ≤ med ≤ max:
16 ≤ 45 ≤ 77
IQR (CV) : 18 (0.3)
48 distinct values 0 (0.0%)
IQ Diff SD [numeric]
Mean (sd) : 6.6 (4.4)
min ≤ med ≤ max:
0.7 ≤ 5.5 ≤ 24.8
IQR (CV) : 5.4 (0.7)
66 distinct values 10 (8.9%)
MD Diff SD [numeric]
Mean (sd) : 8.6 (3.5)
min ≤ med ≤ max:
3.8 ≤ 7.9 ≤ 23.5
IQR (CV) : 4.1 (0.4)
107 distinct values 0 (0.0%)
ERM Diff SD [numeric]
Mean (sd) : 8.6 (2.9)
min ≤ med ≤ max:
3.4 ≤ 8.2 ≤ 17
IQR (CV) : 4.2 (0.3)
101 distinct values 0 (0.0%)
None Diff SD [numeric]
Mean (sd) : 8.7 (2.9)
min ≤ med ≤ max:
3 ≤ 8.4 ≤ 16.4
IQR (CV) : 3.9 (0.3)
107 distinct values 0 (0.0%)
Tested Students [integer]
Mean (sd) : 239.5 (245.9)
min ≤ med ≤ max:
10 ≤ 140.5 ≤ 1009
IQR (CV) : 225.8 (1)
95 distinct values 0 (0.0%)
E% [numeric]
Mean (sd) : 8.6 (12)
min ≤ med ≤ max:
0 ≤ 4 ≤ 60
IQR (CV) : 9.2 (1.4)
29 distinct values 0 (0.0%)
M% [numeric]
Mean (sd) : 30.4 (17.7)
min ≤ med ≤ max:
0 ≤ 32.5 ≤ 71
IQR (CV) : 27.2 (0.6)
50 distinct values 0 (0.0%)
PM% [numeric]
Mean (sd) : 43.6 (16.8)
min ≤ med ≤ max:
0 ≤ 46.5 ≤ 93
IQR (CV) : 20.5 (0.4)
53 distinct values 0 (0.0%)
NM% [numeric]
Mean (sd) : 17.3 (19.1)
min ≤ med ≤ max:
0 ≤ 10 ≤ 89
IQR (CV) : 21.2 (1.1)
44 distinct values 0 (0.0%)
E%State [numeric] 1 distinct value
14 : 112 ( 100.0% )
0 (0.0%)
M%State [numeric] 1 distinct value
36 : 112 ( 100.0% )
0 (0.0%)
PM%State [numeric] 1 distinct value
38 : 112 ( 100.0% )
0 (0.0%)
NM%State [numeric] 1 distinct value
12 : 112 ( 100.0% )
0 (0.0%)
EDiff [numeric]
Mean (sd) : -5.4 (12)
min ≤ med ≤ max:
-14 ≤ -10 ≤ 46
IQR (CV) : 9.2 (-2.2)
29 distinct values 0 (0.0%)
MDiff [numeric]
Mean (sd) : -5.6 (17.7)
min ≤ med ≤ max:
-36 ≤ -3.5 ≤ 35
IQR (CV) : 27.2 (-3.2)
50 distinct values 0 (0.0%)
PMDiff [numeric]
Mean (sd) : 5.6 (16.8)
min ≤ med ≤ max:
-38 ≤ 8.5 ≤ 55
IQR (CV) : 20.5 (3)
53 distinct values 0 (0.0%)
NMDiff [numeric]
Mean (sd) : 5.3 (19.1)
min ≤ med ≤ max:
-12 ≤ -2 ≤ 77
IQR (CV) : 21.2 (3.6)
44 distinct values 0 (0.0%)
EN%State [numeric] 1 distinct value
54 : 112 ( 100.0% )
0 (0.0%)
MF%State [numeric] 1 distinct value
58 : 112 ( 100.0% )
0 (0.0%)
WA%State [numeric] 1 distinct value
51 : 112 ( 100.0% )
0 (0.0%)
IQ%State [numeric] 1 distinct value
55 : 112 ( 100.0% )
0 (0.0%)
MD%State [numeric] 1 distinct value
54 : 112 ( 100.0% )
0 (0.0%)
ERM%State [numeric] 1 distinct value
58 : 112 ( 100.0% )
0 (0.0%)
None%State [numeric] 1 distinct value
51 : 112 ( 100.0% )
0 (0.0%)
School Size [ordered, factor]
1. Small
2. Low-Mid
3. Upper-Mid
4. Large
28 ( 25.0% )
29 ( 25.9% )
27 ( 24.1% )
28 ( 25.0% )
0 (0.0%)
EMDiff [numeric]
Mean (sd) : -10.9 (26.6)
min ≤ med ≤ max:
-50 ≤ -14 ≤ 50
IQR (CV) : 39 (-2.4)
64 distinct values 0 (0.0%)
EM Perf Stat [ordered, factor]
1. HighEM
2. HighM
3. Mid
4. Mid-Low
5. Low
22 ( 19.6% )
16 ( 14.3% )
18 ( 16.1% )
32 ( 28.6% )
24 ( 21.4% )
0 (0.0%)

Generated by summarytools 1.0.1 (R version 4.2.2)
2023-11-19

Key Variables

To explore the relationship between the distribution of school’s students’ Performance Level and school’s performance in content categories, we examine the percentage of points earned by students at schools as well as the standard deviation of the difference between points earned by students at a school and points earned by students in the state across Reporting Categories and Practice Categories. We grouped schools by their EM Perf Stat, an ordinal variable classifying schools by the percentage of students they have that were classified as either Exceeding or Meeting expectations on the MCAS. These numbers seem to suggest that items classified with the Science Practice Category of Mathematics and Data seem to be more challenging to students than those classified as Evidence, Reasoning, and Modeling. These practice categories are strongly and equally emphasized within the exam; items tagged with these categories account for 82% of the available points on the exam with exactly 41% of available points coming from each category.

When considering content Reporting Categories, there do not seem to be discernible distinctions between EM Perf Stat and school’s achievement and performance across categories. All schools seem to perform the strongest on Motion and Forces items, followed by Energy, and weakest on Waves items. Notably, this is also the order of the relative weights of the content areas within the exam; MF, EN, and WA items account for 50%, 30%, and 20% of exam points respectively.

Code 
 #quantile(HSPhy_NextGen_SchoolSum$`EMDiff`)


 
HSPhy_NextGen_SchoolSum%>%
  group_by(`EM Perf Stat`)%>%
    summarise( `Mean MD%` = mean(`MD%`), 
              `Mean MD SD` = mean(`MD Diff SD`),
              `Mean ERM%` = mean(`ERM%`),
               `Mean ERM SD` = mean (`ERM Diff SD`))
Code 
HSPhy_NextGen_SchoolSum%>%
  group_by(`EM Perf Stat`)%>%
    summarise( `Mean MF%` = mean(`MF%`), 
              `Mean MF SD` = mean(`MF Diff SD`),
              `Mean EN%` = mean(`EN%`),
               `Mean EN SD` = mean (`EN Diff SD`),
              `Mean WA%` = mean(`WA%`),
               `Mean WA SD` = mean (`WA Diff SD`)
              )

Visualization

Distribution of Performance Level %

When examining the statewide performance distribution, we can see from the right-skew that it is rare for schools to have high percentages of students classified as Not Meeting expectations and even rarer for schools to have high percentages of students classified as Exceeding expectations.

Code 
HSPhy_NextGen_SchoolSum%>%
  select(`E%`, `M%`, `PM%`, `NM%`)%>%
  pivot_longer(c(1:4), names_to = "Performance Level", values_to = "% Students")%>%
   ggplot( aes(x=`% Students`, color=`Performance Level`, fill=`Performance Level`)) +
    geom_histogram(alpha=0.6, binwidth = 15) +
    scale_fill_viridis(discrete=TRUE) +
    scale_color_viridis(discrete=TRUE) +
    #theme_ipsum() +
    theme(
      legend.position="none",
      panel.spacing = unit(0.1, "lines"),
      strip.text.x = element_text(size = 8)
    ) +
  
    facet_wrap(~`Performance Level`)+
      labs( y = "",
            title = "School Performance Level Distribution",
            x = "% Students at Performance Level",
            caption = "NextGen HS Physics MCAS")

Distribution of School Performance and Variability by Practice Cat

Although Mathematics and Data and Evidence, Reasoning, and Modeling items have strong and equal weighting in the HS Introductory Physics exam, student performance distributions are noticeably different across these practice categories.

Code 
HSPhy_NextGen_SchoolSum%>%
  select(`ERM%`, `MD%`)%>%
  pivot_longer(c(1:2), names_to = "Practice Cat", values_to = "% Points")%>%
   ggplot( aes(x=`% Points`, color=`Practice Cat`, fill=`Practice Cat`)) +
    geom_histogram(alpha=0.6, binwidth = 3) +
    scale_fill_viridis(discrete=TRUE) +
    scale_color_viridis(discrete=TRUE) +
    #theme_ipsum() +
    theme(
      panel.spacing = unit(0.1, "lines"),
      strip.text.x = element_text(size = 8)
    ) +
  
    facet_wrap(~`Practice Cat`)+
      labs( y = "",
            title = "School Performance by Practice Category",
            x = "% Points Earned",
            caption = "NextGen HS Physics MCAS")

Code 
  #ggtitle("Practice Category Performance")

When considering the variability of a school’s performance on items relative to the state by Practice Category, SD MD Diff, and SD ERM Diff, we can see that Mathematics and Data is skewed more to the right.

Code 
  HSPhy_NextGen_SchoolSum%>%
  select(`ERM Diff SD`, `MD Diff SD`)%>%
  pivot_longer(c(1:2), names_to = "Practice Cat", values_to = "SD Diff")%>%
   ggplot( aes(x=`SD Diff`, color=`Practice Cat`, fill=`Practice Cat`)) +
    geom_histogram(alpha=0.6, binwidth = 3) +
    scale_fill_viridis(discrete=TRUE) +
    scale_color_viridis(discrete=TRUE) +
   # theme_ipsum() +
    theme(
      panel.spacing = unit(0.1, "lines"),
      strip.text.x = element_text(size = 8)
    ) +
      labs( y = "",
            title = "School Performance Variation by Practice Category",
            x = "SD Diff",
            caption = "NextGen HS Physics MCAS") +
    facet_wrap(~`Practice Cat`)

Mathematics and Data vs. Evidence Reasoning and Modeling (Practice Category)

These images, seem to suggest that schools with the highest percentage of students classified as Exceeding expectations on the MCAS have the lowest levels of variation in performance on Mathematics and Data Items and schools with the lowest percentage of students classified as Exceeding expectations on the MCAS have the highest levels of variation in performance on Mathematics and Data Items.

Code 
HSPhy_NextGen_SchoolSum%>%
  select(`EM Perf Stat`, `ERM Diff SD`, `MD Diff SD` )%>%
  pivot_longer(c(2:3), names_to = "Practice Cat", values_to = "SD Diff")%>%
  ggplot( aes(x= `EM Perf Stat`, y=`SD Diff`, fill= `EM Perf Stat`)) +
    geom_boxplot() +
    scale_fill_viridis(discrete = TRUE, alpha=0.6) +
    geom_jitter(color="black", size=0.4, alpha=0.9) +
    
    theme(
      plot.title = element_text(size=11),
      axis.title.x=element_blank(),
      #axis.text.x=element_blank()
    ) +
 
    labs( y = "SD Diff",
            title = "Student Performance Variation by Practice Category",
            x = "",
            caption = "NextGen HS Physics MCAS") +
  facet_wrap(~`Practice Cat`)

Code 
HSPhy_NextGen_SchoolSum%>%
  select(`EM Perf Stat`, `ERM Diff SD`, `MD Diff SD` )%>%
  pivot_longer(c(2:3), names_to = "Practice Cat", values_to = "SD Diff")%>%
  ggplot( aes(x= `Practice Cat`, y=`SD Diff`, fill= `Practice Cat`)) +
    geom_boxplot() +
    scale_fill_viridis(discrete = TRUE, alpha=0.6) +
    geom_jitter(color="black", size=0.4, alpha=0.9) +
    #theme_ipsum() +
    theme(
     
      plot.title = element_text(size=11),
      axis.title.x=element_blank(),
      axis.text.x=element_blank()
    ) +
     labs( y = "SD Diff",
            title = "Student Practice Cat. Variation by Achievement Level",
            x = "",
            caption = "NextGen HS Physics MCAS") +
    #xlab("")+
  facet_wrap(~`EM Perf Stat`)

These images, seem to suggest that students at all schools seem to have more difficulty with Mathematics and Data items as compared to Evidence, Reasoning, and Modeling Items.

Code 
HSPhy_NextGen_SchoolSum%>%
  select(`EM Perf Stat`, `ERM%`, `MD%` )%>%
  pivot_longer(c(2:3), names_to = "Practice Cat", values_to = "%Points")%>%
  ggplot( aes(x= `EM Perf Stat`, y=`%Points`, fill= `EM Perf Stat`)) +
    geom_boxplot() +
    scale_fill_viridis(discrete = TRUE, alpha=0.6) +
    geom_jitter(color="black", size=0.4, alpha=0.9) +
    #theme_ipsum() +
    theme(
      
      plot.title = element_text(size=11)
    ) +
    labs( y = "%Points Earned",
            title = "Student Practice Cat. Achievement by Performance Level",
            x = "",
            caption = "NextGen HS Physics MCAS") +
    #xlab("")+
  facet_wrap(~`Practice Cat`)

Code 
HSPhy_NextGen_SchoolSum%>%
  select(`EM Perf Stat`, `ERM%`, `MD%` )%>%
  pivot_longer(c(2:3), names_to = "Practice Cat", values_to = "%Points")%>%
  ggplot( aes(x= `Practice Cat`, y=`%Points`, fill= `Practice Cat`)) +
    geom_boxplot() +
    scale_fill_viridis(discrete = TRUE, alpha=0.6) +
    geom_jitter(color="black", size=0.4, alpha=0.9) +
    #theme_ipsum() +
    theme(
      
      plot.title = element_text(size=11)
    ) +
    labs( y = "%Points Earned",
            title = "Student Practice Cat. Achievement by Performance Level",
            x = "",
            caption = "NextGen HS Physics MCAS") +
    #xlab("")+
  facet_wrap(~`EM Perf Stat`, scale ="free_y")

Code 
# HSPhy_NextGen_SchoolSum%>%
#   select(`EM Perf Stat`, `ERMDiff`, `MDDiff` )%>%
#   pivot_longer(c(2:3), names_to = "Practice Cat", values_to = "%Points")%>%
#   ggplot( aes(x= `EM Perf Stat`, y=`%Points`, fill= `EM Perf Stat`)) +
#     geom_boxplot() +
#     scale_fill_viridis(discrete = TRUE, alpha=0.6) +
#     geom_jitter(color="black", size=0.4, alpha=0.9) +
#     #theme_ipsum() +
#     theme(
#       
#       plot.title = element_text(size=11)
#     ) +
#     labs( y = "%Points Earned",
#             title = "Student Practice Cat. Achievement by Performance Level",
#             x = "",
#             caption = "NextGen HS Physics MCAS") +
#     #xlab("")+
#   facet_wrap(~`Practice Cat`)

# HSPhy_NextGen_SchoolSum%>%
#   select(`EM Perf Stat`, `ERM%`, `MD%` )%>%
#   pivot_longer(c(2:3), names_to = "Practice Cat", values_to = "%Points")%>%
#   ggplot( aes(x= `Practice Cat`, y=`%Points`, fill= `Practice Cat`)) +
#     geom_boxplot() +
#     scale_fill_viridis(discrete = TRUE, alpha=0.6) +
#     geom_jitter(color="black", size=0.4, alpha=0.9) +
#     #theme_ipsum() +
#     theme(
#       
#       plot.title = element_text(size=11)
#     ) +
#     labs( y = "%Points Earned",
#             title = "Student Practice Cat. Achievement by Performance Level",
#             x = "",
#             caption = "NextGen HS Physics MCAS") +
#     #xlab("")+
#   facet_wrap(~`EM Perf Stat`, scale ="free_y")
  

# HSPhy_NextGen_SchoolSum %>%
#   ggplot( aes(x= `EM Perf Stat`, y=`MD%`, fill= `EM Perf Stat`)) +
#     geom_boxplot() +
#     scale_fill_viridis(discrete = TRUE, alpha=0.6) +
#     geom_jitter(color="black", size=0.4, alpha=0.9) +
#     theme_ipsum() +
#     theme(
#       legend.position="none",
#       plot.title = element_text(size=11)
#     ) +
#     ggtitle("A boxplot with jitter") +
#     xlab("")

# HSPhy_NextGen_SchoolSum %>%
#   ggplot( aes(x= `EM Perf Stat`, y=`MD Diff SD`, fill= `EM Perf Stat`)) +
#     geom_boxplot() +
#     scale_fill_viridis(discrete = TRUE, alpha=0.6) +
#     geom_jitter(color="black", size=0.4, alpha=0.9) +
#     theme_ipsum() +
#     theme(
#       legend.position="none",
#       plot.title = element_text(size=11)
#     ) +
#     ggtitle("A boxplot with jitter") +
#     xlab("")
# 
# HSPhy_NextGen_SchoolSum %>%
#   ggplot( aes(x= `EM Perf Stat`, y=`ERM Diff SD`, fill= `EM Perf Stat`)) +
#     geom_boxplot() +
#     scale_fill_viridis(discrete = TRUE, alpha=0.6) +
#     geom_jitter(color="black", size=0.4, alpha=0.9) +
#     theme_ipsum() +
#     theme(
#       legend.position="none",
#       plot.title = element_text(size=11)
#     ) +
#     ggtitle("A boxplot with jitter") +
#     xlab("")
# 
# HSPhy_NextGen_SchoolSum %>%
#   ggplot( aes(x= `EM Perf Stat`, y=`ERM%`, fill= `EM Perf Stat`)) +
#     geom_boxplot() +
#     scale_fill_viridis(discrete = TRUE, alpha=0.6) +
#     geom_jitter(color="black", size=0.4, alpha=0.9) +
#     theme_ipsum() +
#     theme(
#       legend.position="none",
#       plot.title = element_text(size=11)
#     ) +
#     ggtitle("A boxplot with jitter") +
#     xlab("")

Distribution of School Performance and Variability by Reporting Cat

Here we can visualize the variability of a school’s performance on items partitioned by Content Reporting Category of Motion and Forces, Energy, and Waves via: MF%/SD MF Diff, EN%/SD EN Diff, and WA%/SD WA Diff.

Code 
  HSPhy_NextGen_SchoolSum%>%
  select(`EM Perf Stat`, `MF Diff SD`, `EN Diff SD`, `WA Diff SD` )%>%
  pivot_longer(c(2:4), names_to = "Report Cat", values_to = "SD Diff")%>%
  ggplot( aes(x=`SD Diff`, color=`Report Cat`, fill=`Report Cat`)) +
    geom_histogram(alpha=0.6, binwidth = 3) +
    scale_fill_viridis(discrete=TRUE) +
    scale_color_viridis(discrete=TRUE) +
    #theme_ipsum() +
    theme(
      panel.spacing = unit(0.1, "lines"),
      strip.text.x = element_text(size = 8)
    ) +
      labs( y = "",
            title = "School Performance Variation by Content Reporting Category",
            x = "SD Diff",
            caption = "NextGen HS Physics MCAS") +
  facet_wrap(~`Report Cat`)

Code 
HSPhy_NextGen_SchoolSum%>%
  select(`EM Perf Stat`, `MF%`, `EN%`, `WA%` )%>%
  pivot_longer(c(2:4), names_to = "Report Cat", values_to = "% Points")%>%
 ggplot( aes(x=`% Points`, color=`Report Cat`, fill=`Report Cat`)) +
    geom_histogram(alpha=0.6, binwidth = 3) +
    scale_fill_viridis(discrete=TRUE) +
    scale_color_viridis(discrete=TRUE) +
    #theme_ipsum() +
    theme(
      panel.spacing = unit(0.1, "lines"),
      strip.text.x = element_text(size = 8)
    ) +
  
    facet_wrap(~`Report Cat`)+
      labs( y = "",
            title = "Student Performance by Content Reporting Category",
            x = "% Points Earned",
            caption = "NextGen HS Physics MCAS")

Code 
  #ggtitle("Practice Category Performance")

Motion and Forces vs. Energy vs. Waves (Reporting Category)

These images suggest that most schools exhibit similar levels of variability in performance relative to the state across all reporting categories. Schools with the lowest percentage of students Exceeding expectations exhibit high variability in performance across all content reporting categories, but seem to have lower variability on Waves items.

Code 
HSPhy_NextGen_SchoolSum%>%
  select(`EM Perf Stat`, `MF Diff SD`, `EN Diff SD`, `WA Diff SD` )%>%
  pivot_longer(c(2:4), names_to = "Report Cat", values_to = "SD Diff")%>%
  ggplot( aes(x= `EM Perf Stat`, y=`SD Diff`, fill= `EM Perf Stat`)) +
    geom_boxplot() +
    scale_fill_viridis(discrete = TRUE, alpha=0.6) +
    geom_jitter(color="black", size=0.4, alpha=0.9) +
    
    theme(
      plot.title = element_text(size=11),
      axis.title.x=element_blank(),
      axis.text.x=element_blank()
    ) +
 
    labs( y = "SD Diff",
            title = "School Performance Variation by Content Reporting Category",
            x = "",
            caption = "NextGen HS Physics MCAS") +
  facet_wrap(~`Report Cat`)

Code 
HSPhy_NextGen_SchoolSum%>%
  select(`EM Perf Stat`, `MF Diff SD`, `EN Diff SD`, `WA Diff SD` )%>%
  pivot_longer(c(2:4), names_to = "Report Cat", values_to = "SD Diff")%>%
  ggplot( aes(x= `Report Cat`, y=`SD Diff`, fill= `Report Cat`)) +
    geom_boxplot() +
    scale_fill_viridis(discrete = TRUE, alpha=0.6) +
    geom_jitter(color="black", size=0.4, alpha=0.9) +
    #theme_ipsum() +
    theme(
     
      plot.title = element_text(size=11),
      axis.title.x=element_blank(),
      axis.text.x=element_blank()
    ) +
     labs( y = "SD Diff",
            title = "School Content Reporting Cat. Variation by Achievement Level",
            x = "",
            caption = "NextGen HS Physics MCAS") +
    #xlab("")+
  facet_wrap(~`EM Perf Stat`)

Code 
HSPhy_NextGen_SchoolSum%>%
  select(`EM Perf Stat`, `MF%`, `EN%`, `WA%` )%>%
  pivot_longer(c(2:4), names_to = "Report Cat", values_to = "% Points")%>%
  ggplot( aes(x= `Report Cat`, y=`% Points`, fill= `Report Cat`)) +
    geom_boxplot() +
    scale_fill_viridis(discrete = TRUE, alpha=0.6) +
    geom_jitter(color="black", size=0.4, alpha=0.9) +
    #theme_ipsum() +
    theme(

      plot.title = element_text(size=11),
      axis.title.x=element_blank(),
      axis.text.x=element_blank()
    ) +
     labs( y = "Report Cat%",
            title = "School Content Reporting Cat. Performance by Achievement Level",
            x = "",
            caption = "NextGen HS Physics MCAS") +
    #xlab("")+
  facet_wrap(~`EM Perf Stat`)

Code 
# HSPhy_NextGen_SchoolSum %>%
#   ggplot( aes(x= `EM Perf Stat`, y=`MF%`, fill= `EM Perf Stat`)) +
#     geom_boxplot() +
#     scale_fill_viridis(discrete = TRUE, alpha=0.6) +
#     geom_jitter(color="black", size=0.4, alpha=0.9) +
#     theme_ipsum() +
#     theme(
#       legend.position="none",
#       plot.title = element_text(size=11)
#     ) +
#     ggtitle("A boxplot with jitter") +
#     xlab("")
# 
# HSPhy_NextGen_SchoolSum %>%
#   ggplot( aes(x= `EM Perf Stat`, y=`MF Diff SD`, fill= `EM Perf Stat`)) +
#     geom_boxplot() +
#     scale_fill_viridis(discrete = TRUE, alpha=0.6) +
#     geom_jitter(color="black", size=0.4, alpha=0.9) +
#     theme_ipsum() +
#     theme(
#       legend.position="none",
#       plot.title = element_text(size=11)
#     ) +
#     ggtitle("A boxplot with jitter") +
#     xlab("")
# 
# 
# 
# HSPhy_NextGen_SchoolSum %>%
#   ggplot( aes(x= `EM Perf Stat`, y=`EN%`, fill= `EM Perf Stat`)) +
#     geom_boxplot() +
#     scale_fill_viridis(discrete = TRUE, alpha=0.6) +
#     geom_jitter(color="black", size=0.4, alpha=0.9) +
#     theme_ipsum() +
#     theme(
#       legend.position="none",
#       plot.title = element_text(size=11)
#     ) +
#     ggtitle("A boxplot with jitter") +
#     xlab("")
# 
# HSPhy_NextGen_SchoolSum %>%
#   ggplot( aes(x= `EM Perf Stat`, y=`EN Diff SD`, fill= `EM Perf Stat`)) +
#     geom_boxplot() +
#     scale_fill_viridis(discrete = TRUE, alpha=0.6) +
#     geom_jitter(color="black", size=0.4, alpha=0.9) +
#     theme_ipsum() +
#     theme(
#       legend.position="none",
#       plot.title = element_text(size=11)
#     ) +
#     ggtitle("A boxplot with jitter") +
#     xlab("")
# 
# HSPhy_NextGen_SchoolSum %>%
#   ggplot( aes(x= `EM Perf Stat`, y=`WA%`, fill= `EM Perf Stat`)) +
#     geom_boxplot() +
#     scale_fill_viridis(discrete = TRUE, alpha=0.6) +
#     geom_jitter(color="black", size=0.4, alpha=0.9) +
#     theme_ipsum() +
#     theme(
#       legend.position="none",
#       plot.title = element_text(size=11)
#     ) +
#     ggtitle("A boxplot with jitter") +
#     xlab("")
# 
# HSPhy_NextGen_SchoolSum %>%
#   ggplot( aes(x= `EM Perf Stat`, y=`WA Diff SD`, fill= `EM Perf Stat`)) +
#     geom_boxplot() +
#     scale_fill_viridis(discrete = TRUE, alpha=0.6) +
#     geom_jitter(color="black", size=0.4, alpha=0.9) +
#     theme_ipsum() +
#     theme(
#       legend.position="none",
#       plot.title = element_text(size=11)
#     ) +
#     ggtitle("A boxplot with jitter") +
#     xlab("")

Hypothesis Testing

Hypothesis 1: Variation in Mathematics

  • A school’s percentage of students classified as Exceeding or Meeting expectations on the Introductory Physics MCAS is negatively associated with a school’s variance in performance relative to students in the state on Mathematics and Data items, SD MD Diff.
Code 
HSPhy_NextGen_SchoolSum<-HSPhy_NextGen_SchoolSum%>%
  ungroup()

HSPhy_NextGen_SchoolSum<-HSPhy_NextGen_SchoolSum%>%
  mutate(`EorM%` = `E%` + `M%`)

HSPhy_NextGen_SchoolSum

Anova: SD-Diff MD

To explore the relationship between the variance in a schools’ Diff compared to the state on Mathematics and Data items, MD Diff SD and a school’s percentage of students meeting or exceeding expectations on the MCAS, EorM%, we ran a few Hypothesis tests. We considered the impact of the variability in Evidence, Reasoning, and Modeling, ERM Diff SD, and School Size as controls. From our table, there appears to be a relationship between School Size. It appears that Small schools have a higher variation in mathematics and data items and typically perform worse on Mathematics and Data and overall on the MCAS compared to larger schools.

Code 
HSPhy_NextGen_SchoolSum%>%
  group_by(`School Size`)%>%
  summarize(
    `Mean EorM%` = mean(`EorM%`),
      `Mean MD%` = mean(`MD%`),
     `Mean MD Diff SD` = mean(`MD Diff SD`),
     `Mean ERM%` = mean(`ERM%`),
     `Mean ERM Diff SD` = mean(`ERM Diff SD`)
             )
Code 
HSPhy_NextGen_SchoolSum%>%
  select(`School Size`, `MD Diff SD`, `ERM Diff SD` )%>%
  pivot_longer(c(2:3), names_to = "Practice Cat", values_to = "SD Diff")%>%
  ggplot( aes(x= `Practice Cat`, y=`SD Diff`, fill= `Practice Cat`)) +
    geom_boxplot() +
    scale_fill_viridis(discrete = TRUE, alpha=0.6) +
    geom_jitter(color="black", size=0.4, alpha=0.9) +
    #theme_ipsum() +
    theme(

      plot.title = element_text(size=11),
      axis.title.x=element_blank(),
      axis.text.x=element_blank()
    ) +
     labs( y = "SD Diff",
            title = "Student Practice Cat. Variation by School Size",
            x = "",
            caption = "NextGen HS Physics MCAS") +
    #xlab("")+
  facet_wrap(~`School Size`, scale = "free")

Code 
HSPhy_NextGen_SchoolSum%>%
  select(`School Size`, `MD%`, `ERM%` )%>%
  pivot_longer(c(2:3), names_to = "Practice Cat", values_to = "%Points")%>%
  ggplot( aes(x= `Practice Cat`, y=`%Points`, fill= `Practice Cat`)) +
    geom_boxplot() +
    scale_fill_viridis(discrete = TRUE, alpha=0.6) +
    geom_jitter(color="black", size=0.4, alpha=0.9) +
    #theme_ipsum() +
    theme(

      plot.title = element_text(size=11),
      axis.title.x=element_blank(),
      axis.text.x=element_blank()
    ) +
     labs( y = "% Points Earned",
            title = "Student Practice Cat. Achievement by School Size",
            x = "",
            caption = "NextGen HS Physics MCAS") +
    #xlab("")+
  facet_wrap(~`School Size`)

However, when you group the schools by EM Perf Stat, you find that the highest performing, High EM, Small schools have a higher percentage of students meeting or exceeding expectations.

Across all sizes, it seems that the weakest performing schools have more variation in mathematics and data and the strongest performing schools have less variability in Mathematics and Data than in Evidence, Reasoning, and Modeling.

Code 
# Faceted by performance level
  HSPhy_NextGen_SchoolSum%>%
  group_by(`School Size`, `EM Perf Stat`)%>%
  summarize(`Mean EorM%` = mean(`EorM%`),
      `Mean MD%` = mean(`MD%`),
     `Mean MD Diff SD` = mean(`MD Diff SD`),
     `Mean ERM%` = mean(`ERM%`),
     `Mean ERM Diff SD` = mean(`ERM Diff SD`)
             )
Code 
HSPhy_NextGen_SchoolSum%>%
  select(`School Size`, `EorM%`, `MD Diff SD`, `EM Perf Stat` )%>%
  #pivot_longer(c(2:3), names_to = "Practice Cat", values_to = "SD Diff")%>%
  ggplot( aes(x= `School Size`, y=`EorM%`, fill= `School Size`)) +
    geom_boxplot() +
    scale_fill_viridis(discrete = TRUE, alpha=0.6) +
    geom_jitter(color="black", size=0.4, alpha=0.9) +
    #theme_ipsum() +
    theme(
     
      plot.title = element_text(size=11),
      axis.title.x=element_blank(),
      axis.text.x=element_blank()
    ) +
     labs( y = "SD Diff",
            title = "Students Meeting or Exceeding Expectations . Variation by Achievement Level",
            x = "",
            caption = "NextGen HS Physics MCAS") +
    #xlab("")+
  facet_wrap(~`EM Perf Stat`, scales = "free")

Code 
ANOVA_MD_size <- aov(`MD Diff SD` ~ `EM Perf Stat` + `School Size`, data=HSPhy_NextGen_SchoolSum)

ANOVA_MD_interact <- aov(`MD Diff SD` ~ `EM Perf Stat` + `School Size` + `EM Perf Stat` * `School Size`, data=HSPhy_NextGen_SchoolSum)

ANOVA_EorM_MD<- aov(`EorM%` ~ `MD Diff SD` + `School Size`, data=HSPhy_NextGen_SchoolSum)

ANOVA_EorM_interact_MD <- aov(`EorM%` ~ `MD Diff SD` + `School Size` + `MD Diff SD` * `School Size`, data=HSPhy_NextGen_SchoolSum)

summary(ANOVA_MD_size)
                Df Sum Sq Mean Sq F value   Pr(>F)    
`EM Perf Stat`   4  642.5   160.6   44.58  < 2e-16 ***
`School Size`    3  356.6   118.8   32.99 4.58e-15 ***
Residuals      104  374.7     3.6                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Code 
summary(ANOVA_MD_interact)
                             Df Sum Sq Mean Sq F value   Pr(>F)    
`EM Perf Stat`                4  642.5  160.62  43.056  < 2e-16 ***
`School Size`                 3  356.6  118.85  31.860 2.95e-14 ***
`EM Perf Stat`:`School Size` 11   27.7    2.52   0.676    0.758    
Residuals                    93  346.9    3.73                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Code 
summary(ANOVA_EorM_MD)
               Df Sum Sq Mean Sq F value   Pr(>F)    
`MD Diff SD`    1  21623   21623  44.558 1.13e-09 ***
`School Size`   3   4794    1598   3.293   0.0234 *  
Residuals     107  51924     485                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Code 
summary(ANOVA_EorM_interact_MD)
                            Df Sum Sq Mean Sq F value   Pr(>F)    
`MD Diff SD`                 1  21623   21623  46.446 6.32e-10 ***
`School Size`                3   4794    1598   3.433   0.0197 *  
`MD Diff SD`:`School Size`   3   3507    1169   2.511   0.0627 .  
Residuals                  104  48417     466                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Code 
HSPhy_NextGen_SchoolSum%>%
  group_by(`School Size`, `EM Perf Stat`)%>%
  summarize(`Mean MD%` = mean(`MD%`),
     `Mean MD Diff SD` = mean(`MD Diff SD`),
     `Mean ERM%` = mean(`ERM%`),
     `Mean ERM Diff SD` = mean(`ERM Diff SD`)
    
             )
Code 
HSPhy_NextGen_SchoolSum%>%
  group_by(`School Size`)%>%
  summarize(
    `Mean EorM%` = mean(`EorM%`),
      `Mean MD%` = mean(`MD%`),
     `Mean MD Diff SD` = mean(`MD Diff SD`),
     `Mean ERM%` = mean(`ERM%`),
     `Mean ERM Diff SD` = mean(`ERM Diff SD`)
             )
Code 
HSPhy_NextGen_SchoolSum%>%
  group_by(`School Size`, `EM Perf Stat`)%>%
  summarize(`Mean EorM%` = mean(`EorM%`),
      `Mean MD%` = mean(`MD%`),
     `Mean MD Diff SD` = mean(`MD Diff SD`),
     `Mean ERM%` = mean(`ERM%`),
     `Mean ERM Diff SD` = mean(`ERM Diff SD`)
             )
Code 
HSPhy_NextGen_SchoolSum%>%
  filter(`School Size` == "Small")

ANOVA SD-Diff ERM

Code 
ANOVA_ERM <- aov(`ERM Diff SD` ~ `EM Perf Stat`, data=HSPhy_NextGen_SchoolSum)
ANOVA_ERM_Size <- aov(`ERM Diff SD` ~ `EM Perf Stat` + `School Size`, data=HSPhy_NextGen_SchoolSum)
summary(ANOVA_ERM)
                Df Sum Sq Mean Sq F value   Pr(>F)    
`EM Perf Stat`   4  311.8   77.95   13.33 7.58e-09 ***
Residuals      107  625.7    5.85                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Code 
summary(ANOVA_ERM_Size)
                Df Sum Sq Mean Sq F value Pr(>F)    
`EM Perf Stat`   4  311.8   77.95   33.61 <2e-16 ***
`School Size`    3  384.5  128.18   55.28 <2e-16 ***
Residuals      104  241.2    2.32                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Code 
ANOVA_EorM_ERM<- aov(`EorM%` ~ `ERM Diff SD` + `School Size`, data=HSPhy_NextGen_SchoolSum)

ANOVA_EorM_interact_ERM <- aov(`EorM%` ~ `ERM Diff SD` + `School Size` + `ERM Diff SD` * `School Size`, data=HSPhy_NextGen_SchoolSum)

summary(ANOVA_EorM_ERM)
               Df Sum Sq Mean Sq F value   Pr(>F)    
`ERM Diff SD`   1  15815   15815  28.531 5.23e-07 ***
`School Size`   3   3216    1072   1.934    0.128    
Residuals     107  59310     554                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Code 
summary(ANOVA_EorM_interact_ERM)
                             Df Sum Sq Mean Sq F value  Pr(>F)    
`ERM Diff SD`                 1  15815   15815  28.561 5.4e-07 ***
`School Size`                 3   3216    1072   1.936   0.128    
`ERM Diff SD`:`School Size`   3   1722     574   1.037   0.380    
Residuals                   104  57588     554                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

SD-Diff MD + ERM

Code 
ANOVA_EorM_MD_ERM <- aov(`EorM%` ~ `MD Diff SD` + `ERM Diff SD`, data=HSPhy_NextGen_SchoolSum)

summary(ANOVA_EorM_MD_ERM)
               Df Sum Sq Mean Sq F value   Pr(>F)    
`MD Diff SD`    1  21623   21623  41.577 3.19e-09 ***
`ERM Diff SD`   1     31      31   0.059    0.809    
Residuals     109  56688     520                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

T Test EM Diff SD vs. MD Diff SD

Code 
 HSPhy_NextGen_SchoolSum_Diff<-HSPhy_NextGen_SchoolSum %>%
  select(`EM Perf Stat`, `ERM Diff SD`, `MD Diff SD` )%>%
  pivot_longer(c(2:3), names_to = "Practice Cat", values_to = "SD Diff")%>%
   group_by(`Practice Cat`, `EM Perf Stat`)%>%
   summarize(`SD SD Diff` = sd(`SD Diff`, na.rm = TRUE),
             `Mean SD Diff` = mean(`SD Diff`, na.rm = TRUE))%>%
  ungroup()

test1 <- HSPhy_NextGen_SchoolSum_Diff%>%
  select(`Practice Cat`, `SD SD Diff`)
t.test(  test1$`SD SD Diff` ~ test1$`Practice Cat`, paired = TRUE)

    Paired t-test

data:  test1$`SD SD Diff` by test1$`Practice Cat`
t = -0.1841, df = 4, p-value = 0.8629
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 -1.0982164  0.9616354
sample estimates:
mean difference 
    -0.06829049
Code 
test2<-HSPhy_NextGen_SchoolSum %>%
  select(`School Name`, `EM Perf Stat`, `School Size`, `ERM Diff SD`, `MD Diff SD` )%>%
  pivot_longer(c(4:5), names_to = "Practice Cat", values_to = "SD Diff")%>%
  filter(`EM Perf Stat` == "HighEM" | `EM Perf Stat` == "Mid")

## filtered for High Performing Schools
test2
Code 
t.test(test2$`SD Diff`~ test2$`Practice Cat`, paired = TRUE)

    Paired t-test

data:  test2$`SD Diff` by test2$`Practice Cat`
t = 2.7086, df = 39, p-value = 0.009983
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 0.130287 0.898713
sample estimates:
mean difference 
         0.5145
Code 
test3<-HSPhy_NextGen_SchoolSum %>%
  select(`School Name`, `EM Perf Stat`, `School Size`, `ERM Diff SD`, `MD Diff SD` )%>%
  pivot_longer(c(4:5), names_to = "Practice Cat", values_to = "SD Diff")%>%
  filter(`EM Perf Stat` == "Low")

## filtered for High Performing Schools
test3
Code 
t.test(test3$`SD Diff`~ test3$`Practice Cat`, paired = TRUE)

    Paired t-test

data:  test3$`SD Diff` by test3$`Practice Cat`
t = -2.4286, df = 23, p-value = 0.02338
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 -2.6033031 -0.2083636
sample estimates:
mean difference 
      -1.405833

Hypothesis 2: Reporting Cateogy and School Performance

  • A school’s summary performance on items in a given content Reporting Category as measured by MF%, EN%, and WA%, is positively associated with the Reporting Category's weight within the exam.

SD-Diff Reporting Categories

Code 
ANOVA_WA <- aov(`WA Diff SD` ~ `EM Perf Stat`, data=HSPhy_NextGen_SchoolSum)

summary(ANOVA_WA)
                Df Sum Sq Mean Sq F value  Pr(>F)   
`EM Perf Stat`   4  137.6   34.39   4.802 0.00133 **
Residuals      107  766.4    7.16                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Code 
ANOVA_EN <- aov(`EN Diff SD` ~ `EM Perf Stat`, data=HSPhy_NextGen_SchoolSum)

summary(ANOVA_EN)
                Df Sum Sq Mean Sq F value   Pr(>F)    
`EM Perf Stat`   4  446.3  111.57    14.7 1.32e-09 ***
Residuals      107  811.8    7.59                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Code 
ANOVA_MF <- aov(`MF Diff SD` ~ `EM Perf Stat`, data=HSPhy_NextGen_SchoolSum)

summary(ANOVA_MF)
                Df Sum Sq Mean Sq F value   Pr(>F)    
`EM Perf Stat`   4  447.2  111.79   19.59 4.02e-12 ***
Residuals      107  610.5    5.71                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Code 
 HSPhy_NextGen_SchoolSum%>%
  select(`EM Perf Stat`, `MF Diff SD`, `EN Diff SD`, `WA Diff SD` )%>%
  pivot_longer(c(2:4), names_to = "Report Cat", values_to = "SD Diff")%>%
   group_by(`Report Cat`, `EM Perf Stat`)%>%
   summarize(`SD SD Diff` = sd(`SD Diff`, na.rm = TRUE))
Code 
  HSPhy_NextGen_SchoolSum%>%
  select(`EM Perf Stat`, `ERM Diff SD`, `MD Diff SD` )%>%
  pivot_longer(c(2:3), names_to = "Practice Cat", values_to = "SD Diff")%>%
   group_by(`Practice Cat`, `EM Perf Stat`)%>%
   summarize(`SD SD Diff` = sd(`SD Diff`, na.rm = TRUE),
             `Mean SD Diff` = mean(`SD Diff`, na.rm = TRUE))

Visualizations

Code 
ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = `ERM Diff SD`, y = (`EorM%`))) +
  geom_point() +
  geom_smooth(method="lm", se=T)

Code 
ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = `MD Diff SD`, y = (`EorM%`))) +
  geom_point() +
  geom_smooth(method="lm", se=T)

Code 
ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = `MD Diff SD`, y = (`E%`))) +
  geom_point() +
  geom_smooth(method="lm", se=T)

Code 
ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = `ERM Diff SD`, y = (`E%`))) +
  geom_point() +
  geom_smooth(method="lm", se=T)

Code 
ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = (`WA Diff SD`), y = ((`E%`)))) +
  geom_point() +
  geom_smooth(method="lm", se=T)

MD Diff Alone

Code 
fit_md = lm(`EorM%` ~ (`MD Diff SD`), data = HSPhy_NextGen_SchoolSum)
summary(fit_md)

Call:
lm(formula = `EorM%` ~ (`MD Diff SD`), data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-32.575 -18.347  -4.778  15.525  81.212 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)   73.1813     5.6876  12.867  < 2e-16 ***
`MD Diff SD`  -3.9674     0.6127  -6.476 2.73e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 22.71 on 110 degrees of freedom
Multiple R-squared:  0.276, Adjusted R-squared:  0.2694 
F-statistic: 41.94 on 1 and 110 DF,  p-value: 2.726e-09

MD and ERM Diff

This states that MD is significant but ERM is not statistically significant?

Code 
fit_md_erm = lm(`EorM%` ~ (`ERM Diff SD` + `MD Diff SD`), data = HSPhy_NextGen_SchoolSum)
summary(fit_md_erm)

Call:
lm(formula = `EorM%` ~ (`ERM Diff SD` + `MD Diff SD`), data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-32.499 -18.077  -4.584  16.254  81.167 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)    74.0590     6.7612  10.953  < 2e-16 ***
`ERM Diff SD`  -0.3279     1.3517  -0.243  0.80876    
`MD Diff SD`   -3.7413     1.1166  -3.351  0.00111 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 22.81 on 109 degrees of freedom
Multiple R-squared:  0.2764,    Adjusted R-squared:  0.2631 
F-statistic: 20.82 on 2 and 109 DF,  p-value: 2.201e-08

Reporting Category DIFF alone and with interactions

EN

Code 
fit_en = lm(`EorM%` ~ (`EN Diff SD`), data = HSPhy_NextGen_SchoolSum)
summary(fit_en)

Call:
lm(formula = `EorM%` ~ (`EN Diff SD`), data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-33.548 -18.356  -4.079  14.269  80.867 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)   70.0957     6.1134  11.466  < 2e-16 ***
`EN Diff SD`  -3.6402     0.6676  -5.453 3.07e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 23.68 on 110 degrees of freedom
Multiple R-squared:  0.2128,    Adjusted R-squared:  0.2056 
F-statistic: 29.74 on 1 and 110 DF,  p-value: 3.073e-07

MF

Code 
fit_mf = lm(`EorM%` ~ (`MF Diff SD`), data = HSPhy_NextGen_SchoolSum)
summary(fit_mf)

Call:
lm(formula = `EorM%` ~ (`MF Diff SD`), data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-31.602 -19.591  -3.146  13.321  82.578 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)   78.6411     6.4338  12.223  < 2e-16 ***
`MF Diff SD`  -4.5448     0.6968  -6.522 2.18e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 22.66 on 110 degrees of freedom
Multiple R-squared:  0.2789,    Adjusted R-squared:  0.2723 
F-statistic: 42.54 on 1 and 110 DF,  p-value: 2.182e-09

WA

Code 
fit_wa = lm(`EorM%` ~ (`WA Diff SD`), data = HSPhy_NextGen_SchoolSum)
summary(fit_wa)

Call:
lm(formula = `EorM%` ~ (`WA Diff SD`), data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-46.473 -21.430  -1.154  16.703  79.358 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)   62.9282     7.4207   8.480 1.15e-13 ***
`WA Diff SD`  -2.8688     0.8444  -3.397 0.000948 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 25.39 on 110 degrees of freedom
Multiple R-squared:  0.09496,   Adjusted R-squared:  0.08673 
F-statistic: 11.54 on 1 and 110 DF,  p-value: 0.0009481

MF/EN

Code 
fit_mf_en = lm(`EorM%` ~ (`MF Diff SD`) + `EN Diff SD` + `MF Diff SD`*`EN Diff SD`, data = HSPhy_NextGen_SchoolSum)
summary(fit_mf_en)

Call:
lm(formula = `EorM%` ~ (`MF Diff SD`) + `EN Diff SD` + `MF Diff SD` * 
    `EN Diff SD`, data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-31.095 -19.080  -3.607  13.209  82.025 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)                88.5460    15.4514   5.731 9.16e-08 ***
`MF Diff SD`               -5.1276     1.9197  -2.671  0.00873 ** 
`EN Diff SD`               -1.6326     2.1240  -0.769  0.44380    
`MF Diff SD`:`EN Diff SD`   0.1096     0.1584   0.692  0.49069    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 22.81 on 108 degrees of freedom
Multiple R-squared:  0.2829,    Adjusted R-squared:  0.2629 
F-statistic:  14.2 on 3 and 108 DF,  p-value: 7.241e-08

MF/WA

Code 
fit_mf_wa = lm(`EorM%` ~ (`MF Diff SD`) + `WA Diff SD` + `MF Diff SD`*`WA Diff SD`, data = HSPhy_NextGen_SchoolSum)
summary(fit_mf_wa)

Call:
lm(formula = `EorM%` ~ (`MF Diff SD`) + `WA Diff SD` + `MF Diff SD` * 
    `WA Diff SD`, data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-30.132 -18.281  -4.656  13.137  78.167 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)                91.2521    20.3203   4.491 1.79e-05 ***
`MF Diff SD`               -7.6468     2.4499  -3.121  0.00231 ** 
`WA Diff SD`               -0.2917     2.5515  -0.114  0.90919    
`MF Diff SD`:`WA Diff SD`   0.2133     0.2450   0.871  0.38595    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 22.55 on 108 degrees of freedom
Multiple R-squared:  0.299, Adjusted R-squared:  0.2795 
F-statistic: 15.35 on 3 and 108 DF,  p-value: 2.181e-08

Practice Cat Interacting with Reporting Cat

MD/WA

Code 
fit_md_wa = lm(`EorM%` ~ (`MD Diff SD`) + `WA Diff SD` + `MD Diff SD`*`WA Diff SD`, data = HSPhy_NextGen_SchoolSum)
summary(fit_md_wa)

Call:
lm(formula = `EorM%` ~ (`MD Diff SD`) + `WA Diff SD` + `MD Diff SD` * 
    `WA Diff SD`, data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-33.160 -16.523  -5.964  14.101  69.660 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)               104.1191    18.7929   5.540 2.15e-07 ***
`MD Diff SD`              -10.0194     2.5033  -4.002 0.000115 ***
`WA Diff SD`               -1.7848     2.1050  -0.848 0.398373    
`MD Diff SD`:`WA Diff SD`   0.4548     0.2177   2.089 0.039048 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 22.18 on 108 degrees of freedom
Multiple R-squared:  0.3218,    Adjusted R-squared:  0.303 
F-statistic: 17.09 on 3 and 108 DF,  p-value: 3.762e-09

MD/MF

Code 
fit_md_mf = lm(`EorM%` ~ (`MD Diff SD`) + `MF Diff SD` + `MD Diff SD`*`MF Diff SD`, data = HSPhy_NextGen_SchoolSum)
summary(fit_md_mf)

Call:
lm(formula = `EorM%` ~ (`MD Diff SD`) + `MF Diff SD` + `MD Diff SD` * 
    `MF Diff SD`, data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-32.536 -16.951  -4.496  12.527  83.141 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)                92.7357    14.3853   6.447 3.29e-09 ***
`MD Diff SD`               -4.5226     2.6891  -1.682   0.0955 .  
`MF Diff SD`               -3.3337     1.9103  -1.745   0.0838 .  
`MD Diff SD`:`MF Diff SD`   0.1680     0.1427   1.178   0.2415    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 22.57 on 108 degrees of freedom
Multiple R-squared:  0.298, Adjusted R-squared:  0.2785 
F-statistic: 15.28 on 3 and 108 DF,  p-value: 2.35e-08

MD/EN

Code 
fit_md_en = lm(`EorM%` ~ (`MD Diff SD`) + `EN Diff SD` + `MD Diff SD`*`EN Diff SD`, data = HSPhy_NextGen_SchoolSum)
summary(fit_md_en)

Call:
lm(formula = `EorM%` ~ (`MD Diff SD`) + `EN Diff SD` + `MD Diff SD` * 
    `EN Diff SD`, data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-34.020 -17.378  -4.997  12.832  86.410 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)                90.9327    13.8560   6.563 1.89e-09 ***
`MD Diff SD`               -6.4081     2.1149  -3.030  0.00306 ** 
`EN Diff SD`               -1.4283     1.7193  -0.831  0.40797    
`MD Diff SD`:`EN Diff SD`   0.1842     0.1292   1.426  0.15687    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 22.7 on 108 degrees of freedom
Multiple R-squared:  0.2894,    Adjusted R-squared:  0.2697 
F-statistic: 14.66 on 3 and 108 DF,  p-value: 4.462e-08

Practice Cat %

MD/ERM

Code 
fit_md_erm_percent = lm(`EorM%` ~ `MD%` + `ERM%` + `MD%`*`ERM%`, data = HSPhy_NextGen_SchoolSum)
summary(fit_md_erm_percent)

Call:
lm(formula = `EorM%` ~ `MD%` + `ERM%` + `MD%` * `ERM%`, data = HSPhy_NextGen_SchoolSum)

Residuals:
     Min       1Q   Median       3Q      Max 
-18.9971  -2.5815   0.1229   2.3757  16.8436 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -31.643553   5.538802  -5.713 9.92e-08 ***
`MD%`          0.801079   0.206932   3.871 0.000186 ***
`ERM%`         0.148625   0.203262   0.731 0.466240    
`MD%`:`ERM%`   0.009329   0.002118   4.404 2.51e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.473 on 108 degrees of freedom
Multiple R-squared:  0.9724,    Adjusted R-squared:  0.9717 
F-statistic:  1269 on 3 and 108 DF,  p-value: < 2.2e-16

MD/WA

Code 
fit_md_wa_percent = lm(`EorM%` ~ `MD%` + `WA%` + `MD%`*`WA%`, data = HSPhy_NextGen_SchoolSum)
summary(fit_md_wa_percent)

Call:
lm(formula = `EorM%` ~ `MD%` + `WA%` + `MD%` * `WA%`, data = HSPhy_NextGen_SchoolSum)

Residuals:
     Min       1Q   Median       3Q      Max 
-17.1752  -3.0657  -0.1061   2.2054  19.8722 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -35.163765   4.730111  -7.434 2.63e-11 ***
`MD%`         1.290583   0.161619   7.985 1.63e-12 ***
`WA%`        -0.015233   0.205137  -0.074  0.94094    
`MD%`:`WA%`   0.006053   0.002141   2.828  0.00559 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.729 on 108 degrees of freedom
Multiple R-squared:  0.9692,    Adjusted R-squared:  0.9683 
F-statistic:  1132 on 3 and 108 DF,  p-value: < 2.2e-16

ERM/WA

Code 
fit_erm_wa_percent = lm(`EorM%` ~ `ERM%` + `WA%` + `ERM%`*`WA%`, data = HSPhy_NextGen_SchoolSum)
summary(fit_erm_wa_percent)

Call:
lm(formula = `EorM%` ~ `ERM%` + `WA%` + `ERM%` * `WA%`, data = HSPhy_NextGen_SchoolSum)

Residuals:
     Min       1Q   Median       3Q      Max 
-16.8575  -3.3910   0.0816   2.9314  13.5817 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -46.744066   6.471302  -7.223 7.53e-11 ***
`ERM%`         0.993748   0.175196   5.672 1.19e-07 ***
`WA%`          0.315195   0.227940   1.383    0.170    
`ERM%`:`WA%`   0.007465   0.002715   2.749    0.007 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.197 on 108 degrees of freedom
Multiple R-squared:  0.9628,    Adjusted R-squared:  0.9617 
F-statistic:   931 on 3 and 108 DF,  p-value: < 2.2e-16

MD, ERM, MF

Code 
fit_md_erm_mf_percent = lm(`EorM%` ~ `ERM%` + `MD%` + `MF%` + `MD%`*`ERM%`+ `MD%`*`MF%` + `ERM%`*`MF%`, data = HSPhy_NextGen_SchoolSum)
summary(fit_md_erm_mf_percent)

Call:
lm(formula = `EorM%` ~ `ERM%` + `MD%` + `MF%` + `MD%` * `ERM%` + 
    `MD%` * `MF%` + `ERM%` * `MF%`, data = HSPhy_NextGen_SchoolSum)

Residuals:
     Min       1Q   Median       3Q      Max 
-14.4908  -2.3554   0.2093   1.9293  14.2689 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -19.38914    9.12900  -2.124 0.036026 *  
`ERM%`         1.37435    0.84576   1.625 0.107162    
`MD%`          4.55083    0.89029   5.112 1.44e-06 ***
`MF%`         -5.12934    1.11268  -4.610 1.14e-05 ***
`ERM%`:`MD%`  -0.07223    0.02178  -3.317 0.001251 ** 
`MD%`:`MF%`    0.01931    0.01667   1.158 0.249291    
`ERM%`:`MF%`   0.06161    0.01704   3.616 0.000461 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.083 on 105 degrees of freedom
Multiple R-squared:  0.9777,    Adjusted R-squared:  0.9764 
F-statistic: 765.9 on 6 and 105 DF,  p-value: < 2.2e-16

MD, ERM, WA

Code 
fit_md_erm_wa_percent = lm(`EorM%` ~ `ERM%` + `MD%` + `WA%` + `MD%`*`ERM%`+ `MD%`*`WA%` + `ERM%`*`WA%`, data = HSPhy_NextGen_SchoolSum)
summary(fit_md_erm_wa_percent)

Call:
lm(formula = `EorM%` ~ `ERM%` + `MD%` + `WA%` + `MD%` * `ERM%` + 
    `MD%` * `WA%` + `ERM%` * `WA%`, data = HSPhy_NextGen_SchoolSum)

Residuals:
     Min       1Q   Median       3Q      Max 
-15.1584  -2.1387   0.2441   2.1956  14.8754 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -27.02886    8.91137  -3.033  0.00305 ** 
`ERM%`        -0.79907    0.62833  -1.272  0.20628    
`MD%`         -0.73390    0.89506  -0.820  0.41410    
`WA%`          2.44743    0.76083   3.217  0.00172 ** 
`ERM%`:`MD%`   0.05509    0.01285   4.285 4.06e-05 ***
`MD%`:`WA%`   -0.02285    0.01265  -1.806  0.07385 .  
`ERM%`:`WA%`  -0.02437    0.02027  -1.202  0.23192    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.187 on 105 degrees of freedom
Multiple R-squared:  0.9765,    Adjusted R-squared:  0.9752 
F-statistic: 727.3 on 6 and 105 DF,  p-value: < 2.2e-16

MD-Diff, WA Diff, and School Size

Code 
fit_md_wa_size = lm(`EorM%` ~ `MD Diff SD` + `WA Diff SD` + `School Size` + `MD Diff SD`*`School Size`, data = HSPhy_NextGen_SchoolSum)
summary(fit_md_wa_size)

Call:
lm(formula = `EorM%` ~ `MD Diff SD` + `WA Diff SD` + `School Size` + 
    `MD Diff SD` * `School Size`, data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-40.307 -13.762  -3.628  11.108  67.074 

Coefficients:
                             Estimate Std. Error t value Pr(>|t|)    
(Intercept)                   72.1484     9.9200   7.273 7.10e-11 ***
`MD Diff SD`                  -7.1651     1.2880  -5.563 2.11e-07 ***
`WA Diff SD`                   3.1782     1.2298   2.584   0.0112 *  
`School Size`.L               13.4575    16.2678   0.827   0.4100    
`School Size`.Q              -35.9805    17.2147  -2.090   0.0391 *  
`School Size`.C               20.4199    18.6382   1.096   0.2758    
`MD Diff SD`:`School Size`.L  -0.3028     2.2505  -0.135   0.8932    
`MD Diff SD`:`School Size`.Q   5.4903     2.2503   2.440   0.0164 *  
`MD Diff SD`:`School Size`.C  -1.8429     2.2897  -0.805   0.4228    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 21.01 on 103 degrees of freedom
Multiple R-squared:  0.4196,    Adjusted R-squared:  0.3745 
F-statistic: 9.308 on 8 and 103 DF,  p-value: 1.385e-09
Code 
HSPhy_NextGen_SchoolSum

Scatter Plots

MD + WA

Code 
fit_md_wa$coefficients
              (Intercept)              `MD Diff SD`              `WA Diff SD` 
               104.119115                -10.019434                 -1.784838 
`MD Diff SD`:`WA Diff SD` 
                 0.454809
Code 
summary(fit_md_wa)

Call:
lm(formula = `EorM%` ~ (`MD Diff SD`) + `WA Diff SD` + `MD Diff SD` * 
    `WA Diff SD`, data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-33.160 -16.523  -5.964  14.101  69.660 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)               104.1191    18.7929   5.540 2.15e-07 ***
`MD Diff SD`              -10.0194     2.5033  -4.002 0.000115 ***
`WA Diff SD`               -1.7848     2.1050  -0.848 0.398373    
`MD Diff SD`:`WA Diff SD`   0.4548     0.2177   2.089 0.039048 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 22.18 on 108 degrees of freedom
Multiple R-squared:  0.3218,    Adjusted R-squared:  0.303 
F-statistic: 17.09 on 3 and 108 DF,  p-value: 3.762e-09
Code 
ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = -10.019434*`MD Diff SD` +  -1.7848*`WA Diff SD` + 0.4548*`MD Diff SD`*`WA Diff SD` + 104.1191, y = `EorM%`)) +
  geom_point() +
  geom_smooth(method="lm", se=T)

Code 
ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = `MD Diff SD` +  `WA Diff SD`, y = `EorM%`)) +
  geom_point() +
  geom_smooth(method="lm", se=T)

Code 
ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = 1.290583*`MD%` +  -0.015233*`WA%` +  0.006053*`WA%`*`MD%` + -35.163765 , y = `EorM%`)) +
  geom_point() +
  geom_smooth(method="lm", se=T)

Code 
#fit_erm = lm(`E%` ~ `ERM Diff SD`, data = HSPhy_NextGen_SchoolSum)
#summary(fit_erm)

#fit_erm_md = lm(`E%` ~ log(`MD Diff SD`) + log(`ERM Diff SD`) + log(`MD Diff SD`)*log(`ERM Diff SD`), data = HSPhy_NextGen_SchoolSum)
#summary(fit_erm_md)


#fit_md_percent = lm(`E%` ~ log(`MD%`) + log(`ERM%`) + log(`MD%`)*log(`ERM%`), data = HSPhy_NextGen_SchoolSum)

#summary(fit_md_percent)

HSPhy_NextGen_SchoolSum%>%
  select(`MD%`, `E%`)
Code 
ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = `MD%`, y = log(`E%`))) +
  geom_point() +
  geom_smooth(method="lm", se=T)

Code 
ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = `ERM%`, y = log(`E%`))) +
  geom_point() +
  geom_smooth(method="lm", se=T)

Code 
ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = log(`MD Diff SD`), y = (`E%`))) +
  geom_point() +
  geom_smooth(method="lm", se=T)

Code 
#HSPhy_NextGen_SchoolSum.isna()

fit_wa = lm((`E%`) ~ log(`WA%`), data = HSPhy_NextGen_SchoolSum)
summary(fit_wa)

Call:
lm(formula = (`E%`) ~ log(`WA%`), data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-10.575  -5.428  -2.218   4.028  33.155 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -104.236      9.087  -11.47   <2e-16 ***
log(`WA%`)    29.999      2.408   12.46   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.761 on 110 degrees of freedom
Multiple R-squared:  0.5853,    Adjusted R-squared:  0.5816 
F-statistic: 155.3 on 1 and 110 DF,  p-value: < 2.2e-16
Code 
fit_md = lm((`E%`) ~ log(`MD%`), data = HSPhy_NextGen_SchoolSum)
summary(fit_md)

Call:
lm(formula = (`E%`) ~ log(`MD%`), data = HSPhy_NextGen_SchoolSum)

Residuals:
    Min      1Q  Median      3Q     Max 
-11.069  -5.617  -1.981   3.093  35.895 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -93.347      8.951  -10.43   <2e-16 ***
log(`MD%`)    26.803      2.344   11.43   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 8.147 on 110 degrees of freedom
Multiple R-squared:  0.5431,    Adjusted R-squared:  0.5389 
F-statistic: 130.7 on 1 and 110 DF,  p-value: < 2.2e-16
Code 
#fit_erm = lm(`E%` ~ `ERM Diff SD`, data = HSPhy_NextGen_SchoolSum)
#summary(fit_erm)

#fit_erm_md = lm(`E%` ~ log(`MD Diff SD`) + log(`ERM Diff SD`) + log(`MD Diff SD`)*log(`ERM Diff SD`), data = HSPhy_NextGen_SchoolSum)
#summary(fit_erm_md)


#fit_md_percent = lm(`E%` ~ log(`MD%`) + log(`ERM%`) + log(`MD%`)*log(`ERM%`), data = HSPhy_NextGen_SchoolSum)

#summary(fit_md_percent)

#HSPhy_NextGen_SchoolSum%>%
#  select(`MD%`, `E%`)

ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = log(`WA%`), y = log(`E%`))) +
  geom_point() +
  geom_smooth(method="lm", se=T)

Code 
#ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = `ERM%`, y = log(`E%`))) +
 # geom_point() +
#  geom_smooth(method="lm", se=T)

#ggplot(data = HSPhy_NextGen_SchoolSum, aes(x = log(`MD Diff SD`), y = (`E%`))) +
 # geom_point() +
#  geom_smooth(method="lm", se=T)

References

Chief State School Officers, Council of. 2021. “How States Can Support the Adoption & Effective Use of High Quality Standards-Aligned Instructional Materials.” https://753a0706.flowpaper.com/CCSSOIMPDCaseStudyOVERVIEWFINAL/#page=1.
Chingos, & Whitehurst, M. M. 2012. “Choosing Blindly. Instructional Materials, Teacher Effectiveness, and the Common Core.” https://www.brookings.edu/wp-content/uploads/2016/06/0410_curriculum_chingos_whitehurst.pdf.
DESE. 2022. “HighSchool Introductory Physics Item Report.” https://profiles.doe.mass.edu/mcas/mcasitems2.aspx?grade=HS&subjectcode=PHY&linkid=23&orgcode=04830000&fycode=2022&orgtypecode=5&.
Greer, W. 2018. “The 50 Year History of the Common Core.” The Journal of Educational Foundations 31 (3&4): 100–117. https://files.eric.ed.gov/fulltext/EJ1212104.pdf.
Papy, Mantil, J. P. 2020. “Lifting All Boats? Accomplishments and Challenges from 20 Years of Education Reform in Massachusetts.” https://annenberg.brown.edu/sites/default/files/LiftingAllBoats_FINAL.pdf.